1.1 --- a/doc-src/TutorialI/CTL/PDL.thy Tue Oct 03 22:39:49 2000 +0200
1.2 +++ b/doc-src/TutorialI/CTL/PDL.thy Wed Oct 04 17:35:45 2000 +0200
1.3 @@ -9,73 +9,179 @@
1.4 (syntax) trees, they are naturally modelled as a datatype:
1.5 *}
1.6
1.7 -datatype pdl_form = Atom atom
1.8 - | NOT pdl_form
1.9 - | And pdl_form pdl_form
1.10 - | AX pdl_form
1.11 - | EF pdl_form;
1.12 +datatype formula = Atom atom
1.13 + | Neg formula
1.14 + | And formula formula
1.15 + | AX formula
1.16 + | EF formula
1.17
1.18 text{*\noindent
1.19 +This is almost the same as in the boolean expression case study in
1.20 +\S\ref{sec:boolex}, except that what used to be called @{text Var} is now
1.21 +called @{term Atom}.
1.22 +
1.23 The meaning of these formulae is given by saying which formula is true in
1.24 which state:
1.25 *}
1.26
1.27 -consts valid :: "state \<Rightarrow> pdl_form \<Rightarrow> bool" ("(_ \<Turnstile> _)" [80,80] 80)
1.28 +consts valid :: "state \<Rightarrow> formula \<Rightarrow> bool" ("(_ \<Turnstile> _)" [80,80] 80)
1.29 +
1.30 +text{*\noindent
1.31 +The concrete syntax annotation allows us to write @{term"s \<Turnstile> f"} instead of
1.32 +@{text"valid s f"}.
1.33 +
1.34 +The definition of @{text"\<Turnstile>"} is by recursion over the syntax:
1.35 +*}
1.36
1.37 primrec
1.38 "s \<Turnstile> Atom a = (a \<in> L s)"
1.39 -"s \<Turnstile> NOT f = (\<not>(s \<Turnstile> f))"
1.40 +"s \<Turnstile> Neg f = (\<not>(s \<Turnstile> f))"
1.41 "s \<Turnstile> And f g = (s \<Turnstile> f \<and> s \<Turnstile> g)"
1.42 "s \<Turnstile> AX f = (\<forall>t. (s,t) \<in> M \<longrightarrow> t \<Turnstile> f)"
1.43 "s \<Turnstile> EF f = (\<exists>t. (s,t) \<in> M^* \<and> t \<Turnstile> f)";
1.44
1.45 -text{*
1.46 +text{*\noindent
1.47 +The first three equations should be self-explanatory. The temporal formula
1.48 +@{term"AX f"} means that @{term f} is true in all next states whereas
1.49 +@{term"EF f"} means that there exists some future state in which @{term f} is
1.50 +true. The future is expressed via @{text"^*"}, the transitive reflexive
1.51 +closure. Because of reflexivity, the future includes the present.
1.52 +
1.53 Now we come to the model checker itself. It maps a formula into the set of
1.54 -states where the formula is true and is defined by recursion over the syntax:
1.55 +states where the formula is true and is defined by recursion over the syntax,
1.56 +too:
1.57 *}
1.58
1.59 -consts mc :: "pdl_form \<Rightarrow> state set";
1.60 +consts mc :: "formula \<Rightarrow> state set";
1.61 primrec
1.62 "mc(Atom a) = {s. a \<in> L s}"
1.63 -"mc(NOT f) = -mc f"
1.64 +"mc(Neg f) = -mc f"
1.65 "mc(And f g) = mc f \<inter> mc g"
1.66 "mc(AX f) = {s. \<forall>t. (s,t) \<in> M \<longrightarrow> t \<in> mc f}"
1.67 "mc(EF f) = lfp(\<lambda>T. mc f \<union> M^-1 ^^ T)"
1.68
1.69
1.70 -text{*
1.71 -Only the equation for @{term EF} deserves a comment: the postfix @{text"^-1"}
1.72 -and the infix @{text"^^"} are predefined and denote the converse of a
1.73 -relation and the application of a relation to a set. Thus @{term "M^-1 ^^ T"}
1.74 -is the set of all predecessors of @{term T} and @{term"mc(EF f)"} is the least
1.75 -set @{term T} containing @{term"mc f"} and all predecessors of @{term T}.
1.76 +text{*\noindent
1.77 +Only the equation for @{term EF} deserves some comments. Remember that the
1.78 +postfix @{text"^-1"} and the infix @{text"^^"} are predefined and denote the
1.79 +converse of a relation and the application of a relation to a set. Thus
1.80 +@{term "M^-1 ^^ T"} is the set of all predecessors of @{term T} and the least
1.81 +fixpoint (@{term lfp}) of @{term"\<lambda>T. mc f \<union> M^-1 ^^ T"} is the least set
1.82 +@{term T} containing @{term"mc f"} and all predecessors of @{term T}. If you
1.83 +find it hard to see that @{term"mc(EF f)"} contains exactly those states from
1.84 +which there is a path to a state where @{term f} is true, do not worry---that
1.85 +will be proved in a moment.
1.86 +
1.87 +First we prove monotonicity of the function inside @{term lfp}
1.88 *}
1.89
1.90 -lemma mono_lemma: "mono(\<lambda>T. A \<union> M^-1 ^^ T)"
1.91 -apply(rule monoI);
1.92 -by(blast);
1.93 +lemma mono_ef: "mono(\<lambda>T. A \<union> M^-1 ^^ T)"
1.94 +apply(rule monoI)
1.95 +by(blast)
1.96
1.97 -lemma lfp_conv_EF:
1.98 -"lfp(\<lambda>T. A \<union> M^-1 ^^ T) = {s. \<exists>t. (s,t) \<in> M^* \<and> t \<in> A}";
1.99 +text{*\noindent
1.100 +in order to make sure it really has a least fixpoint.
1.101 +
1.102 +Now we can relate model checking and semantics. For the @{text EF} case we need
1.103 +a separate lemma:
1.104 +*}
1.105 +
1.106 +lemma EF_lemma:
1.107 + "lfp(\<lambda>T. A \<union> M^-1 ^^ T) = {s. \<exists>t. (s,t) \<in> M^* \<and> t \<in> A}"
1.108 +
1.109 +txt{*\noindent
1.110 +The equality is proved in the canonical fashion by proving that each set
1.111 +contains the other; the containment is shown pointwise:
1.112 +*}
1.113 +
1.114 apply(rule equalityI);
1.115 apply(rule subsetI);
1.116 - apply(simp);
1.117 - apply(erule Lfp.induct);
1.118 - apply(rule mono_lemma);
1.119 - apply(simp);
1.120 + apply(simp)
1.121 +(*pr(latex xsymbols symbols);*)
1.122 +txt{*\noindent
1.123 +Simplification leaves us with the following first subgoal
1.124 +\begin{isabelle}
1.125 +\ \isadigit{1}{\isachardot}\ {\isasymAnd}s{\isachardot}\ s\ {\isasymin}\ lfp\ {\isacharparenleft}{\isasymlambda}T{\isachardot}\ A\ {\isasymunion}\ M{\isacharcircum}{\isacharminus}\isadigit{1}\ {\isacharcircum}{\isacharcircum}\ T{\isacharparenright}\ {\isasymLongrightarrow}\ {\isasymexists}t{\isachardot}\ {\isacharparenleft}s{\isacharcomma}\ t{\isacharparenright}\ {\isasymin}\ M{\isacharcircum}{\isacharasterisk}\ {\isasymand}\ t\ {\isasymin}\ A
1.126 +\end{isabelle}
1.127 +which is proved by @{term lfp}-induction:
1.128 +*}
1.129 +
1.130 + apply(erule Lfp.induct)
1.131 + apply(rule mono_ef)
1.132 + apply(simp)
1.133 +(*pr(latex xsymbols symbols);*)
1.134 +txt{*\noindent
1.135 +Having disposed of the monotonicity subgoal,
1.136 +simplification leaves us with the following main goal
1.137 +\begin{isabelle}
1.138 +\ \isadigit{1}{\isachardot}\ {\isasymAnd}s{\isachardot}\ s\ {\isasymin}\ A\ {\isasymor}\isanewline
1.139 +\ \ \ \ \ \ \ \ \ s\ {\isasymin}\ M{\isacharcircum}{\isacharminus}\isadigit{1}\ {\isacharcircum}{\isacharcircum}\ {\isacharparenleft}lfp\ {\isacharparenleft}{\dots}{\isacharparenright}\ {\isasyminter}\ {\isacharbraceleft}x{\isachardot}\ {\isasymexists}t{\isachardot}\ {\isacharparenleft}x{\isacharcomma}\ t{\isacharparenright}\ {\isasymin}\ M{\isacharcircum}{\isacharasterisk}\ {\isasymand}\ t\ {\isasymin}\ A{\isacharbraceright}{\isacharparenright}\isanewline
1.140 +\ \ \ \ \ \ \ \ \ {\isasymLongrightarrow}\ {\isasymexists}t{\isachardot}\ {\isacharparenleft}s{\isacharcomma}\ t{\isacharparenright}\ {\isasymin}\ M{\isacharcircum}{\isacharasterisk}\ {\isasymand}\ t\ {\isasymin}\ A
1.141 +\end{isabelle}
1.142 +which is proved by @{text blast} with the help of a few lemmas about
1.143 +@{text"^*"}:
1.144 +*}
1.145 +
1.146 apply(blast intro: r_into_rtrancl rtrancl_trans);
1.147 -apply(rule subsetI);
1.148 -apply(simp);
1.149 -apply(erule exE);
1.150 -apply(erule conjE);
1.151 -apply(erule_tac P = "t\<in>A" in rev_mp);
1.152 -apply(erule converse_rtrancl_induct);
1.153 - apply(rule ssubst[OF lfp_Tarski[OF mono_lemma]]);
1.154 - apply(blast);
1.155 -apply(rule ssubst[OF lfp_Tarski[OF mono_lemma]]);
1.156 -by(blast);
1.157 +
1.158 +txt{*
1.159 +We now return to the second set containment subgoal, which is again proved
1.160 +pointwise:
1.161 +*}
1.162 +
1.163 +apply(rule subsetI)
1.164 +apply(simp, clarify)
1.165 +(*pr(latex xsymbols symbols);*)
1.166 +txt{*\noindent
1.167 +After simplification and clarification we are left with
1.168 +\begin{isabelle}
1.169 +\ \isadigit{1}{\isachardot}\ {\isasymAnd}s\ t{\isachardot}\ {\isasymlbrakk}{\isacharparenleft}s{\isacharcomma}\ t{\isacharparenright}\ {\isasymin}\ M{\isacharcircum}{\isacharasterisk}{\isacharsemicolon}\ t\ {\isasymin}\ A{\isasymrbrakk}\ {\isasymLongrightarrow}\ s\ {\isasymin}\ lfp\ {\isacharparenleft}{\isasymlambda}T{\isachardot}\ A\ {\isasymunion}\ M{\isacharcircum}{\isacharminus}\isadigit{1}\ {\isacharcircum}{\isacharcircum}\ T{\isacharparenright}
1.170 +\end{isabelle}
1.171 +This goal is proved by induction on @{term"(s,t)\<in>M^*"}. But since the model
1.172 +checker works backwards (from @{term t} to @{term s}), we cannot use the
1.173 +induction theorem @{thm[source]rtrancl_induct} because it works in the
1.174 +forward direction. Fortunately the converse induction theorem
1.175 +@{thm[source]converse_rtrancl_induct} already exists:
1.176 +@{thm[display,margin=60]converse_rtrancl_induct[no_vars]}
1.177 +It says that if @{prop"(a,b):r^*"} and we know @{prop"P b"} then we can infer
1.178 +@{prop"P a"} provided each step backwards from a predecessor @{term z} of
1.179 +@{term b} preserves @{term P}.
1.180 +*}
1.181 +
1.182 +apply(erule converse_rtrancl_induct)
1.183 +(*pr(latex xsymbols symbols);*)
1.184 +txt{*\noindent
1.185 +The base case
1.186 +\begin{isabelle}
1.187 +\ \isadigit{1}{\isachardot}\ {\isasymAnd}t{\isachardot}\ t\ {\isasymin}\ A\ {\isasymLongrightarrow}\ t\ {\isasymin}\ lfp\ {\isacharparenleft}{\isasymlambda}T{\isachardot}\ A\ {\isasymunion}\ M{\isacharcircum}{\isacharminus}\isadigit{1}\ {\isacharcircum}{\isacharcircum}\ T{\isacharparenright}
1.188 +\end{isabelle}
1.189 +is solved by unrolling @{term lfp} once
1.190 +*}
1.191 +
1.192 + apply(rule ssubst[OF lfp_Tarski[OF mono_ef]])
1.193 +(*pr(latex xsymbols symbols);*)
1.194 +txt{*
1.195 +\begin{isabelle}
1.196 +\ \isadigit{1}{\isachardot}\ {\isasymAnd}t{\isachardot}\ t\ {\isasymin}\ A\ {\isasymLongrightarrow}\ t\ {\isasymin}\ A\ {\isasymunion}\ M{\isacharcircum}{\isacharminus}\isadigit{1}\ {\isacharcircum}{\isacharcircum}\ lfp\ {\isacharparenleft}{\isasymlambda}T{\isachardot}\ A\ {\isasymunion}\ M{\isacharcircum}{\isacharminus}\isadigit{1}\ {\isacharcircum}{\isacharcircum}\ T{\isacharparenright}
1.197 +\end{isabelle}
1.198 +and disposing of the resulting trivial subgoal automatically:
1.199 +*}
1.200 +
1.201 + apply(blast)
1.202 +
1.203 +txt{*\noindent
1.204 +The proof of the induction step is identical to the one for the base case:
1.205 +*}
1.206 +
1.207 +apply(rule ssubst[OF lfp_Tarski[OF mono_ef]])
1.208 +by(blast)
1.209 +
1.210 +text{*
1.211 +The main theorem is proved in the familiar manner: induction followed by
1.212 +@{text auto} augmented with the lemma as a simplification rule.
1.213 +*}
1.214
1.215 theorem "mc f = {s. s \<Turnstile> f}";
1.216 apply(induct_tac f);
1.217 -by(auto simp add:lfp_conv_EF);
1.218 +by(auto simp add:EF_lemma);
1.219 (*<*)end(*>*)